The expectation, or expected value, of a random variable is simply its probability-weighted average value ($\mu_x$). In the discrete case, the expected value is $E(X) = \displaystyle \sum_{x}^\infty x * P(X=x))$ In the continuous case, the expected value is $E(X) = \int\limits_{-\infty}^{\infty}x \ f(x) \ dx$ where $f(x)$ is the [[probability density function]]. ## Properties of expectations If c is constant, then $E(c) = c$. If $a$ and $c$ are constants and $X$ is a random variable, then $E(aX + c) = aE(x) + c$ Expectation is a linear operator, in other words $E(aX + bY + c) = aE(x) + bE(Y) +c$ If $g(X)$ is any function of $X$, then $E(g(X)) = \sum_{x=1}^\infty g(x) P(X=x))$This is described more by the [[Law of the Unthinking Statistician]]. ## deriving expectation of a discrete variable Continuing the example from deriving the [[probability mass function]], let's derive the expectation of the [[first success distribution]]. Recall our pmf is defined as $P(X=x) = (1-p)^{x-1} * p$ We can plug this into our equation for expectation. $E(X) = \sum_{x=1}^\infty x * p(1-p)^{x-1}$ Recall from the [[geometric series]] that $\sum_{k=1}^\infty ar^{k-1} = \frac{a}{1-r} \text {, where r<=1}$ We can find the [[derivative]] of both sides with respect to $r$ to find $\sum_{k=1}^\infty a(k-1)r^{k-2} = \frac{a}{(1-r)^2}$ Observing that when $k=1$, the left side of the equation will be $0$, we can find simply step up the sum to begin from $k=2$. $\sum_{k=2}^\infty a(k-1)r^{k-2} = \frac{a}{(1-r)^2}$ We can then reindex with $k-1 = j$ to get $\sum_{j=1}^\infty ajr^{j-1} = \frac{a}{(1-r)^2}$ Notice that the left side of the equation is now the exact same form as the equation for expectation with $a=p$ and $r = 1-p$, so we can substitute the right side of the equation in for expectation. $E(X) = \sum_{x=1}^\infty x * p(1-p)^{x-1} = \frac{a}{(1-r)^2} = \frac{p}{(1-(1-p))^2} = \frac1p$ This was quite a bit of work to work out the expectation of a discrete random variable! In practice, expectations for specific distributions can be found using lookup tables for common distributions.